The goal of this set of practice problems is to practice the following skills:
Building appropriate multiple linear regression models to address research questions
Interpreting linear regression model results with a mix of quantitative and categorical predictors
Directions
Create a code chunk in which you load the ggplot2, dplyr, and readr packages. Include the following command in the code chunk to read in the data: lifts <- read_csv("https://mac-stat.github.io/data/powerlifting.csv")
Continue with the exercises below. You will need to create new code chunks to construct visualizations and models and write interpretations beneath. Put text responses in blockquotes as shown below:
Response here. (The > at the start of the line starts a blockquote and makes the text larger and easier to read.)
Render your work for submission:
Click the “Render” button in the menu bar for this pane (blue arrow pointing right). This will create an HTML file containing all of the directions, code, and responses from this activity. A preview of the HTML will appear in the browser.
Scroll through and inspect the document to check that your work translated to the HTML format correctly.
Close the browser tab.
Go to the “Background Jobs” pane in RStudio and click the Stop button to end the rendering process.
Locate the rendered HTML file in the folder where this file is saved. Open the HTML to ensure that your work looks as it should (code appears, output displays, interpretations appear). Upload this HTML file to Moodle.
Exercises
Context
Powerlifting is a sport in which athletes compete to lift as much as possible in 3 events: bench press, squat, and deadlift.
Open Powerlifting maintains a database of competition results for powerlifters across the world. We have information on 100,000 lifters from this database. Take a look at the codebook here.
Research question: Are lighter or heavier lifters proportionately stronger?
# Load packages and import datalibrary(readr)library(ggplot2)library(dplyr)
Attaching package: 'dplyr'
The following objects are masked from 'package:stats':
filter, lag
The following objects are masked from 'package:base':
intersect, setdiff, setequal, union
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (10): Name, Sex, Event, Equipment, Place, Tested, Country, State, MeetC...
dbl (10): Age, BodyweightKg, Best3SquatKg, Best3BenchKg, Best3DeadliftKg, T...
date (1): Date
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
Exercise 1: Define outcome variable
Use the mutate() function from the dplyr to define an outcome variable called SWR that stands for strength-to-weight ratio. It should be computed as TotalKg divided by BodyweightKg. SWR measures how many times their bodyweight an athlete can lift in total. Higher numbers indicate higher relative strength.
# A tibble: 6 × 22
Name Sex Event Equipment Age BodyweightKg Best3SquatKg Best3BenchKg
<chr> <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl>
1 Natalya Po… F D Raw 37 58.4 NA NA
2 Fatima Rod… F SBD Single-p… NA 74.8 NA NA
3 Josh Kelley M SBD Single-p… NA 72.4 147. 97.5
4 Timothy Ca… M D Raw 16 72.9 NA NA
5 M Moynihan M B Raw NA 67.5 NA 100
6 Lucas Wegr… M B Raw 23.5 103. NA 188.
# ℹ 14 more variables: Best3DeadliftKg <dbl>, TotalKg <dbl>, Place <chr>,
# Dots <dbl>, Wilks <dbl>, Glossbrenner <dbl>, Goodlift <dbl>, Tested <chr>,
# Country <chr>, State <chr>, Date <date>, MeetCountry <chr>,
# MeetState <chr>, SWR <dbl>
Exercise 2: Exploratory visualizations
Guiding question: How are age, sex, bodyweight, and equipment usage related to strength?
Construct one visualization for each of these 4 explanatory variables and SWR. For each, write 1-2 sentences summarizing what you learn from the plot. Be sure to discuss trend, variability/dispersion about the trend, and any notable outliers.
# Agelifts %>%ggplot(aes(x = Age, y = SWR)) +geom_point() +geom_smooth(method ="lm", se =FALSE)
`geom_smooth()` using formula = 'y ~ x'
Warning: Removed 47630 rows containing non-finite outside the scale range
(`stat_smooth()`).
Warning: Removed 47630 rows containing missing values or values outside the scale range
(`geom_point()`).
There is a weak downward trend in SWR as age increases. There is a lot of variability in SWR for younger lifters, but less variability for older lifters. There are a few outliers of older lifters with high SWR.
# Sexlifts %>%ggplot(aes(x = Sex, y = SWR)) +geom_boxplot()
Warning: Removed 8752 rows containing non-finite outside the scale range
(`stat_boxplot()`).
There is a difference in SWR between M, F, and other. Notably, M has the highest median SWR and other has the lowest. There is more variability in SWR for M than for F and other. There are a few outliers of M lifters with very high SWR. There are a lot of outliers of F lifters with high SWR.
# Bodyweightlifts %>%ggplot(aes(x = BodyweightKg, y = SWR)) +geom_point() +geom_smooth(method ="lm", se =FALSE)
`geom_smooth()` using formula = 'y ~ x'
Warning: Removed 8752 rows containing non-finite outside the scale range
(`stat_smooth()`).
Warning: Removed 8752 rows containing missing values or values outside the scale range
(`geom_point()`).
There is a weak downward trend in SWR as bodyweight increases. There is a lot of variability in SWR for lighter lifters, but less variability for heavier lifters. There are a few outliers of heavier lifters with low SWR.
# Equipment usagelifts %>%ggplot(aes(x = Equipment, y = SWR)) +geom_boxplot()
Warning: Removed 8752 rows containing non-finite outside the scale range
(`stat_boxplot()`).
There is a difference in SWR between the different equipment categories. There is more variability in SWR for multi-ply and raw lifters than for other categories. There is little variability in SWR for straps lifters. There are a many outliers of single-ply and unlimited lifters with high SWR. There are a lot of outliers of wraps lifters with either very high or very low SWR.
Exercise 3: Causal diagram
We are interested in the relationship between BodyweightKg and SWR but are concerned about Age, Sex, and Equipment as potential confounders.
Part a
Draw a causal diagram that shows how these 5 variables might be related. Draw this by hand or software and save the file as pp3_dag.jpg or pp3_dag.png in the same folder as this .qmd file. You can then insert the diagram as below:
Part b
Use visualizations to explore if Age, Sex, and Equipment have a relationship with BodyweightKg. Explain how these explorations relate to your causal diagram. Do you think that there are other confounders that would be important to consider but are missing from our data?
# Agelifts %>%ggplot(aes(x = Age, y = BodyweightKg)) +geom_point() +geom_smooth(method ="lm", se =FALSE)
`geom_smooth()` using formula = 'y ~ x'
Warning: Removed 44123 rows containing non-finite outside the scale range
(`stat_smooth()`).
Warning: Removed 44123 rows containing missing values or values outside the scale range
(`geom_point()`).
There is a weak positive trend in bodyweight as age increases. This is consistent with the causal diagram, as age is a cause of bodyweight.
# Sexlifts %>%ggplot(aes(x = Sex, y = BodyweightKg)) +geom_boxplot()
Warning: Removed 1590 rows containing non-finite outside the scale range
(`stat_boxplot()`).
There is a difference in bodyweight between M, F, and other. Notably, M has the highest median bodyweight and F has the lowest. This is consistent with the causal diagram. There is more variability in bodyweight for M than for F and other. There are a few outliers of M lifters with very high bodyweight. There are a lot of outliers of F lifters with high bodyweight.
# Equipment usagelifts %>%ggplot(aes(x = Equipment, y = BodyweightKg)) +geom_boxplot()
Warning: Removed 1590 rows containing non-finite outside the scale range
(`stat_boxplot()`).
There is minimal difference in bodyweight between the different equipment categories. This is consistent with the causal diagram, as equipment does not effect bodyweight.
There are likely other confounders that would be important to consider but are missing from our data, such as diet, training, and genetics.
Exercise 4: Linear regression modeling
Research question: Are lighter or heavier lifters proportionately stronger?
Put another way, this question is getting at the causal effect of bodyweight on SWR.
Part a
Fit an appropriate linear regression model that answers our research question.
model <-lm(SWR ~ BodyweightKg + Age + Sex + Equipment, data = lifts)summary(model)
Call:
lm(formula = SWR ~ BodyweightKg + Age + Sex + Equipment, data = lifts)
Residuals:
Min 1Q Median 3Q Max
-5.1199 -1.9295 0.2902 1.6151 8.4393
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.8000828 0.0708945 67.707 < 2e-16 ***
BodyweightKg -0.0074596 0.0004803 -15.532 < 2e-16 ***
Age -0.0306179 0.0007904 -38.736 < 2e-16 ***
SexM 1.0120370 0.0224800 45.020 < 2e-16 ***
SexMx -0.0450155 0.8429823 -0.053 0.957413
EquipmentRaw 0.0328689 0.0577330 0.569 0.569137
EquipmentSingle-ply 0.6371731 0.0607246 10.493 < 2e-16 ***
EquipmentStraps -0.7145558 2.0651714 -0.346 0.729342
EquipmentUnlimited -0.6251146 0.1681247 -3.718 0.000201 ***
EquipmentWraps 1.5674820 0.0640535 24.471 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.064 on 52360 degrees of freedom
(47630 observations deleted due to missingness)
Multiple R-squared: 0.1146, Adjusted R-squared: 0.1145
F-statistic: 753.1 on 9 and 52360 DF, p-value: < 2.2e-16
Part b
Interpret the coefficient that answers our research question. Make sure to use appropriate causation vs. association language, include units, and talk about averages rather than individual cases.
The coefficient for BodyweightKg is -0.007. This means that, on average, for every 1 kg increase in bodyweight, the SWR decreases by 0.007. This suggests that heavier lifters are proportionately weaker than lighter lifters. This is a causal interpretation, as we have controlled for confounders in our model.
Part c
Interpret the remainder of the coefficients (including the intercept). Is it meaningful to interpret the intercept in this context?
The intercept is 4.80. This means that, on average, F lifters with multi-ply and a bodyweight of 0 kg has a SWR of 1.5. This is not meaningful in this context, as a lifter with a bodyweight of 0 kg is not possible.
The coefficient for Age is -0.031, which means that, on average, for every 1 year increase in age, the SWR decreases by 0.031.
The coefficient for SexM is 0.101, which means that, on average, M lifters have a SWR that is 0.101 higher than F lifters.
The coefficient for SexMx is -0.045, which means that, on average, other lifters have a SWR that is 0.045 lower than F lifters.
The coefficient for EquipmentRaw is 0.033, which means that, on average, raw lifters have a SWR that is 0.033 higher than multi-ply lifters.
The coefficient for EquipmentSingle-ply is 0.637, which means that, on average, single-ply lifters have a SWR that is 0.637 higher than multi-ply lifters.
The coefficient for EquipmentStraps is -0.714, which means that, on average, straps lifters have a SWR that is 0.714 lower than multi-ply lifters.
The coefficient for EquipmentUnlimited is -0.625, which means that, on average, unlimited lifters have a SWR that is 0.625 lower than multi-ply lifters.
The coefficient for EquipmentWraps is 1.567, which means that, on average, wraps lifters have a SWR that is 1.567 higher than multi-ply lifters.
Source Code
---title: "Practice Problems 3"author: "David Trinh"date: nowdate-format: "YYYY-MM-DDTHH:mm:ssZ"format: html: embed-resources: true code-tools: true ---<center>**Due Friday, 10/4 at 5pm on [Moodle](INSERT_LINK_TO_MOODLE_ASSIGNMENT).**</center># PurposeThe goal of this set of practice problems is to practice the following skills:- Building appropriate multiple linear regression models to address research questions- Interpreting linear regression model results with a mix of quantitative and categorical predictors# Directions1. Create a code chunk in which you load the `ggplot2`, `dplyr`, and `readr` packages. Include the following command in the code chunk to read in the data: `lifts <- read_csv("https://mac-stat.github.io/data/powerlifting.csv")`2. Continue with the exercises below. You will need to create new code chunks to construct visualizations and models and write interpretations beneath. Put text responses in blockquotes as shown below:> Response here. (The > at the start of the line starts a blockquote and makes the text larger and easier to read.)3. Render your work for submission: - Click the "Render" button in the menu bar for this pane (blue arrow pointing right). This will create an HTML file containing all of the directions, code, and responses from this activity. A preview of the HTML will appear in the browser. - Scroll through and inspect the document to check that your work translated to the HTML format correctly. - Close the browser tab. - Go to the "Background Jobs" pane in RStudio and click the Stop button to end the rendering process. - Locate the rendered HTML file in the folder where this file is saved. Open the HTML to ensure that your work looks as it should (code appears, output displays, interpretations appear). Upload this HTML file to Moodle.# Exercises## ContextPowerlifting is a sport in which athletes compete to lift as much as possible in 3 events: bench press, squat, and deadlift.[Open Powerlifting](https://www.openpowerlifting.org/) maintains a database of competition results for powerlifters across the world. We have information on 100,000 lifters from this database. Take a look at the codebook [here](https://github.com/Mac-STAT/data/blob/main/powerlifting_codebook.md).Research question: Are lighter or heavier lifters proportionately stronger?```{r}# Load packages and import datalibrary(readr)library(ggplot2)library(dplyr)lifts <-read_csv("https://mac-stat.github.io/data/powerlifting.csv")```## Exercise 1: Define outcome variableUse the `mutate()` function from the `dplyr` to define an outcome variable called `SWR` that stands for strength-to-weight ratio. It should be computed as `TotalKg` divided by `BodyweightKg`. `SWR` measures how many times their bodyweight an athlete can lift in total. Higher numbers indicate higher relative strength.```{r}lifts <- lifts %>%mutate(SWR = TotalKg / BodyweightKg)head(lifts)```## Exercise 2: Exploratory visualizations**Guiding question:** How are age, sex, bodyweight, and equipment usage related to strength?Construct one visualization for each of these 4 explanatory variables and `SWR`. For each, write 1-2 sentences summarizing what you learn from the plot. Be sure to discuss trend, variability/dispersion about the trend, and any notable outliers.```{r}# Agelifts %>%ggplot(aes(x = Age, y = SWR)) +geom_point() +geom_smooth(method ="lm", se =FALSE)```> There is a weak downward trend in SWR as age increases. There is a lot of variability in SWR for younger lifters, but less variability for older lifters. There are a few outliers of older lifters with high SWR.```{r}# Sexlifts %>%ggplot(aes(x = Sex, y = SWR)) +geom_boxplot()```> There is a difference in SWR between M, F, and other. Notably, M has the highest median SWR and other has the lowest. There is more variability in SWR for M than for F and other. There are a few outliers of M lifters with very high SWR. There are a lot of outliers of F lifters with high SWR.```{r}# Bodyweightlifts %>%ggplot(aes(x = BodyweightKg, y = SWR)) +geom_point() +geom_smooth(method ="lm", se =FALSE)```> There is a weak downward trend in SWR as bodyweight increases. There is a lot of variability in SWR for lighter lifters, but less variability for heavier lifters. There are a few outliers of heavier lifters with low SWR.```{r}# Equipment usagelifts %>%ggplot(aes(x = Equipment, y = SWR)) +geom_boxplot()```> There is a difference in SWR between the different equipment categories. There is more variability in SWR for multi-ply and raw lifters than for other categories. There is little variability in SWR for straps lifters. There are a many outliers of single-ply and unlimited lifters with high SWR. There are a lot of outliers of wraps lifters with either very high or very low SWR.## Exercise 3: Causal diagramWe are interested in the relationship between `BodyweightKg` and `SWR` but are concerned about `Age`, `Sex`, and `Equipment` as potential confounders.### Part aDraw a causal diagram that shows how these 5 variables might be related. Draw this by hand or software and save the file as `pp3_dag.jpg` or `pp3_dag.png` in the same folder as this `.qmd` file. You can then insert the diagram as below:### Part bUse visualizations to explore if `Age`, `Sex`, and `Equipment` have a relationship with `BodyweightKg`. Explain how these explorations relate to your causal diagram. Do you think that there are other confounders that would be important to consider but are missing from our data?```{r}# Agelifts %>%ggplot(aes(x = Age, y = BodyweightKg)) +geom_point() +geom_smooth(method ="lm", se =FALSE)```> There is a weak positive trend in bodyweight as age increases. This is consistent with the causal diagram, as age is a cause of bodyweight.```{r}# Sexlifts %>%ggplot(aes(x = Sex, y = BodyweightKg)) +geom_boxplot()```> There is a difference in bodyweight between M, F, and other. Notably, M has the highest median bodyweight and F has the lowest. This is consistent with the causal diagram. There is more variability in bodyweight for M than for F and other. There are a few outliers of M lifters with very high bodyweight. There are a lot of outliers of F lifters with high bodyweight.```{r}# Equipment usagelifts %>%ggplot(aes(x = Equipment, y = BodyweightKg)) +geom_boxplot()```> There is minimal difference in bodyweight between the different equipment categories. This is consistent with the causal diagram, as equipment does not effect bodyweight.> There are likely other confounders that would be important to consider but are missing from our data, such as diet, training, and genetics.## Exercise 4: Linear regression modelingResearch question: Are lighter or heavier lifters proportionately stronger?Put another way, this question is getting at the causal effect of bodyweight on SWR.### Part aFit an appropriate linear regression model that answers our research question.```{r}model <-lm(SWR ~ BodyweightKg + Age + Sex + Equipment, data = lifts)summary(model)```### Part bInterpret the coefficient that answers our research question. *Make sure to use appropriate causation vs. association language, include units, and talk about averages rather than individual cases.*> The coefficient for `BodyweightKg` is -0.007. This means that, on average, for every 1 kg increase in bodyweight, the SWR decreases by 0.007. This suggests that heavier lifters are proportionately weaker than lighter lifters. This is a causal interpretation, as we have controlled for confounders in our model.### Part cInterpret the remainder of the coefficients (including the intercept). Is it meaningful to interpret the intercept in this context?> The intercept is 4.80. This means that, on average, F lifters with multi-ply and a bodyweight of 0 kg has a SWR of 1.5. This is not meaningful in this context, as a lifter with a bodyweight of 0 kg is not possible.> The coefficient for `Age` is -0.031, which means that, on average, for every 1 year increase in age, the SWR decreases by 0.031.> The coefficient for `SexM` is 0.101, which means that, on average, M lifters have a SWR that is 0.101 higher than F lifters.> The coefficient for `SexMx` is -0.045, which means that, on average, other lifters have a SWR that is 0.045 lower than F lifters.> The coefficient for `EquipmentRaw` is 0.033, which means that, on average, raw lifters have a SWR that is 0.033 higher than multi-ply lifters.> The coefficient for `EquipmentSingle-ply` is 0.637, which means that, on average, single-ply lifters have a SWR that is 0.637 higher than multi-ply lifters.> The coefficient for `EquipmentStraps` is -0.714, which means that, on average, straps lifters have a SWR that is 0.714 lower than multi-ply lifters.> The coefficient for `EquipmentUnlimited` is -0.625, which means that, on average, unlimited lifters have a SWR that is 0.625 lower than multi-ply lifters.> The coefficient for `EquipmentWraps` is 1.567, which means that, on average, wraps lifters have a SWR that is 1.567 higher than multi-ply lifters.
